Review of Lar Felker's book
"The Evans Equations of Unified Field Theory" Rev.3.2

Quotes from Felker's book in black, comments in blue

p.14

⁴ As this book was being written, it was hard to keep up with developments. For example, explanation of the Faraday homopolar generator which has been a mystery for 130 years has been published.

The author should have a look into the relevant physics text books. Then he will detect that the mysteries of the Faraday homopolar generator are his own mysteries only.

p.26

a × b is called the cross product. If a and b are 2-dimensional vectors, the cross product gives a resulting vector in a third dimension

"Perpendicular to both 3-D vectors a and b" would be correct, i.e. both a and b and the resulting vector a×b are 3-D vectors.

Generalizing to 4 dimensions, if A and B are vectors in the xyz-volume, then vector C = A × B will be perpendicular or “orthogonal” to the xyz-volume.

WRONG: A vector C = A × B is not defined in 4-D space. See p.341 Cross product

The wedge product is the four dimensional cross product.

The wedge product AÙB in 4-D space is no vector.

p.35

Imagine that we take a penny and a truck, we take them out into a region with very low gravity, and we attach a rocket to each with 1 kg of propellant. We light the propellant. Which – penny or truck – will be going faster when the rocket stops firing? We are applying the same force to each. The penny. At the end of say 1 minute when the rockets go out, the penny will have traveled farther and be going faster. It has less mass and could be accelerated more. If instead we drop each of them from a height of 100 meters, ignoring effects of air resistance, they will both hit the ground at the same time. Gravity applies the same force to each, as did the rocket. Thus, gravity seems like a force but is not a force like the rocket force. It is a false force. It is actually due to curved spacetime. Any object – penny or truck – will follow the curved spacetime at the same rate. Each will accelerate at the same rate. If gravity were a force, different masses would accelerate at different rates.

An erroneous example: The author should apply his "theory" to 1 penny compared with a bag of  1.000.000 pennies (instead of the truck).

p.47

The tetrad is introduced here and will be developed further in later chapters. The tetrad is a 4×4 matrix of 16 components that are built from vectors in the base manifold and the index as shown in Figure 2-9.

NO! A tetrad is a "4-bein" (tetra = 4), i.e. a set of 4 vectors e₀,e₁,e₂,e₃ in the tangent space of each point of the spacetime manifold with certain additional geometric properties. See the remarks to p.49.

p.49

It is built from four 4-vectors in the base manifold - our real universe - and four 4-vectors in the mathematical tangent space.

No, both sets of vectors belong to the tangent space. One set is given by the tangent vectors of the coordinate axes (usually denoted by ∂₀,∂₁,∂₂,∂₃, which have the disadvantage of not being orthonormal relative to the metric in general) the other set (the tetrad) consists of 4 vectors e₀,e₁,e₂,e₃, orthonormal relative to the metric. Since both sets belong to the same linear space there exist linear representations of one system by the other one.

e_a = q_a^μ ∂_μ   and conversely   ∂_μ = q_μ^a e_a .

The coefficients q_a^μ and q_μ^a of these representations can be called "tetrad coefficients", not being identical with the tetrad itself. Both systems of vectors can be used as a reference basis. Each vector V in the tangent space can be represented relative to both sets of reference.

V = V^μ ∂_μ = V^a e_a .

The coordinates of a vector V, V^μ and V^a respectively, relative to the two reference systems are linearly interrelated with the tetrad coefficients as coefficients:

V^a = V^μ q_μ^a   and conversely   V^μ = V^a q_a^μ .

p.51

Four vectors make up what we call a "4-vector".

No! Each element of a 4-dimensional linear vector space is a "4-vector." That is a result of Felker's permanent misunderstanding of the vector concept. A vector

V = V^μ ∂_μ = V^a e_a .

has different four coordinates (not "vectors") V^μ (μ=0,1,2,3) and V^a (a=0,1,2,3) relative to different frames. Felker erroneously considers these coordinates as "vectors" itself; hence he can state «Four vectors make up what we call a "4-vector».

Which is the meaning of the η's in (3)? On p.52 we learn:
A four-vector is multiplied by the factors –1, 1, 1, 1 to find the metric four-vector.

e_{a metric} = η_a e_a (without summation over the tetrad index a)

where η₀=−1 and η_a=+1 for a=1,2,3. Completely misleading.

p.61

ψ(r,t) is the wave function of a particle defined over r space and t time. If squared it becomes a probability. In quantum theory ψ² is used to find the probability of an energy, a position, a time, an angular momentum.

ψ(r,t) as solution of the Schroedinger equation is complex-valued in general. How can it be that ψ² is a probability, i.e. real-valued?

p.83

Figure 4-4
The dot or scalar product is a scalar, just a distance without direction. It can be used to get work or other values also.

The scalar product yields no distance in general. The Figure 4-4 is misleading nonsense.

p.84

Two four-vectors define a four dimensional space and the product of two is in the space.

How that? They (A and B) define a two-dimensional space, the space of all of their linear combinations aA + bB, with the coefficients a,b.
Which product? The scalar or dot product would yield scalars (i.e.no 4-D vectors), the cross product is not defined and the wedge product yields no vectors, and also the outer product (see p. 89-90) doesn't fit. So what?

If an inner product is defined for every point of a space on a tangent space, all the inner products are called the Riemann metric.

OOOOOH!!! How awful! Do whatever you like, however, don't write a book on a topic you have not understood.

p.85

The inner product of two tetrads is: q_μν = q^a_μ q^b_ν η_ab. This is a tensor, the symmetric metric, g_μν, which is the distance between two points or events in 4-dimensional Riemann space – the space of our universe.

What is the inner product of TWO tetrads. From where do we obtain two (different?) tetrads? And the symmetric metric should be the distance between two points or events. What a nonsense!

If three vectors are multiplied, the result is a volume in three dimensions. See Figure 4-5.

WRONG! Not for the double cross-product a×(b×c) as displayed on p.86 Figure 4-6.

p.86

Figure 4-6
The scalar or mixed product is a volume.

OK, but that volume is not given by the double cross-product a × (b × c) displayed in that figure.

p.87

Divergence
The divergence of a vector tells the amount of flow through a point.

So calculate div (a×r) where a is constant and r denotes the radius vector.
Is there any flow of  F=a×r "through a point" or not?

p.95

Evans uses this to decompose the tetrad into parts. Those parts define
gravitation and electromagnetism. He uses a purely mathematical function that
is well known and generalizes to the physical process.
     q_μ^a =     q_μ^{a (S)}     +      q_μ^{a (A)}
           = (symmetric) + (antisymmetric)
           = gravitation   +  electromagnetism
           = curvature    +      torsion
           = distance      +       turning

WRONG!!! The matrix of the tetrad coefficient doesn't transform symmetrically relative to the two indices μ and a. Thus the well known splitting of a matrix Q (cf. p.295: Antisymmetric tensor)
into its symmetric part           Q(S) = ½ [Q + Q^T]
and its antisymmetric part      Q(A) = ½ [Q − Q^T]
where Q^T denotes the transposed matrix of Q, does not apply here. The reason is that both matrices Q and Q^T have different transformation behavior (co- and contravariantly with respect to the index μ), and hence the addition of both matrices doesn't yield a quantity with physical meaning (apples and pears cannot be added), while the lines
           = gravitation   +  electromagnetism
           = curvature    +      torsion
           = distance      +      turning
contain physical/geometrical quantities. Cf. the chart on p.175.

p.96

The Tetrad q^a_μ
. . . . . . . .
In the tetrad matrix each element is the product of two vectors. One vector is in the base manifold and one in the index. q⁰₀ indicates q⁰ times q₀.

Which "vectors" could be meant by q⁰ and q₀ ? Felker cannot distinguish between vector and vector coordinate (see the remarks to p.51). Here "vector" means "vector coordinate" q⁰ or q₀ respectively. So we have an outer product of two vectors in the sense of p.89/90 under consideration yielding the tetrad matrix (q^a_μ). However, such a matrix cannot be invertible since having parallel line vectors, in contradiction to the necessary invertibility of the tetrad matrices (see the remarks to p.49).

Felker's claim of an outer product decomposition of the tetrad matrix cannot be true.

p.97

Let V^a be the four-vector in the tangent orthonormal flat space and V^μ be the corresponding base four-vector, then:

V^a = q^a_μ V^μ                                                                 (7)

q^a_μ is then the tetrad matrix. Here a and μ stand for 16 vectors as shown in the matrix above. Each element of q^a_μ, the tetrad, is V^a V^μ.

The sentence "Here a and μ stand for 16 vectors as shown in the matrix above" means that the tetrad matrices are 4×4 and therefore have 16 numbers (not vectors) as entries.

May we write the last sentence as the equation q^a_μ = V^a V^μ or what? Then the author should explain how that could be deduced from Equ.(7); i.e. from its written out version

V^a = q^a₀ V⁰ + q^a₁ V¹ + q^a₂ V² + q^a₃ V³

where the indices written in italics refer to the coordinate basis of rhe tangent space. Until we get the author's reply we declare the sentence "Each element of q^a_μ, the tetrad, is V^a V^μ" to be nonsense. And he should explain what is meant by "a and μ stand for 16 vectors".

Here is only one vector, namely V, which has 4 coordinates V⁰,V¹,V²,V³ relative to the coordinate basis ∂₀,∂₁,∂₂,∂₃ and other 4 coordinates V⁰,V¹,V²,V³ relative to the tetrad basis e₀,e₁,e₂,e₃:

V = V^μ∂_μ = V^ae_a .

The two different vector coordinate sets are interrelated by the equations (7).

p.109

Figure 4-18 Vector Multiplication

contains several enigmatic and misleading formulations: Which is the meaning of the geometrical symbols?
E.g.: What is meant by

↑ · ↑ · η = distance ?

Perhaps V^aW^bη_ab for two vectors ↑ = V = V^ae_a, ↑ = W = W^ae_a and the Minkowski matrix η = diag(−1, 1, 1, 1) ?
However, that is no distance at all; even in the case V = W we would obtain the square of the Minkowskian length of V, namely the value

− V^{0 2;} + V^{1 2;} + V^{2 2;} + V^{3 2;}

And what shall the reader learn from

Hodge Dual *A = dual of A
Dual of a 3-vector is a plane. ?

p.137/138

The most general asymmetric metric tensor is defined by the outer or tensor product of two tetrads: Here the tetrad is equivalent to metric four-vectors

q^ab_μν = q^a_μ q^b_ν ,     q^ab_μν = q^ab_μν^(S) + q^ab_μν^(A)                                 (2)

An asymmetric form has both symmetry and antisymmetry within it.

None of the above quantities can be identical with the metric tensor g_μν of spacetime as they have two additional upper tetrad indices a,b, which are missing at g_μν. The correct relation is

g_μν = η_ab q^a_μ q^b_ν.

Here, due to the symmetry of (η_ab) = diag(−1 , 1, 1, 1) the matrix g_μν turns out to be symmetric.

Thus, a (nontrivial) antisymmetric part of the metric g_μν does not exist.

The anti-symmetric metric tensor is q^ab_μν^(A). This defines an area, dA, with rotational potential. The antisymmetric metric tensor is defined by the wedge product of two tetrads:

q^ab_μν^(A) = q^a_μÙ q^b_ν                                                 (3)

Symmetry indicates centralized potentials – spherical shapes.
Anti-symmetry always involves rotational potentials – the helix.
Asymmetry indicates both are contained in the same shape.

p.144

The Evans solution, R^a_μ – ½ Rq^a_μ = kT^a_μ, is more fundamental than Einstein’s equation, G_μν = R_μν − ½ g_μν R = − kT_μν.

p.146

The electromagnetic and weak fields are described by a postulate similar to Einstein’s R = kT:

A^a_μ = A⁽⁰⁾ q^a_μ                                                 (12)

A⁽⁰⁾ is a fundamental electromagnetic potential in volts-seconds per meter.

cf. Remarks to p.175

p.151/152

Field Descriptions

The Evans Lemma gives quantum field/matter theory from general relativity. Gravitation is described by the Lemma when the field is the tetrad, q^a_μ.

The other three fields – electromagnetism, weak, and strong - are described when the field is the tetrad multiplied by an appropriate scaling factor and is in the appropriate representation space. For example, the fundamental electromagnetic field has the antisymmetry of equation (6) and is described by equation (12).

Equation (6) on p.140 is

g^ab_μν ^(A) = q^a_{μ ^} q^b_ν                                                 (6)

Evans felt remembered of the antisymmetry of the electromagnetic fundamental tensor F_μν = − F_νμ

The strong and weak fields are described respectively as:

S^a_μ = S(0) q^a_μ                                                 (13)

W^a_μ = W(0) q^a_μ                                                 (14)

Electromagnetism is defined by the torsion form. It is spinning of spacetime itself, not an object imposed upon the spacetime. The weak field is also a torsion form. It is related to electromagnetism. Thus in the neutron's conversion to a proton the torsion form will be involved. We see an electron leave the neutron and a proton remains. Now it is more clear that there was an electrical interaction that has an explanation. See Chapter 12 on the electroweak theory.

That all may be or not: Pure speculations. see p.175

p.174

Of great interest is that the symmetries of the matrix in mathematics are applied to physical quantities:

q^a_μ = q^a_μ^(S) + q^a_μ^(A)

The splitting is coordinate dependent, hence "physically not meaningful".

Gravitation can be symmetric or antisymmetric, however it is always curving.

Electromagnetism can be symmetric or antisymmetric, however it is always turning.

Summary

The wave equation of general relativity and unified field theory is:

(o + kT) q^a_μ = 0

By substituting appropriate representations for the tetrad, the various equations of physics can be derived. Gravitation, electromagnetism, the weak force, and the strong force can all be represented. This simple looking equation can be expanded as given in the chart at the end of the chapter. The wave equation was derived from general relativity using differential geometry.

The factorization of the symmetric and antisymmetric metrics from the asymmetric tetrad is basic differential geometry. It gives four forces – a discovery opening new insights into physics. This is q^a_μ = q^a_μ^(S) + q^a_μ^(A).

The splitting is coordinate dependent, hence "physically not meaningful".

Applied to physics, four potential fields are represented as shown in the chart.

The standard model is not generally covariant – it does not allow calculations of interactions among particles in different gravitational fields, say near a black hole. The standard model does not allow for electromagnetism's and gravitation's mutual effects to be defined.

p.175

The Evans unified equations allow both these processes to be accomplished.

The use of the mathematical representation space for the tangent spacetime “a” is of the essence. This is the Palatini variation of gravitational theory. The tetrad can now be expressed as any of the four fields of physics – G, A, W, and S. Thus four forms of energy can be described within the unified field q^a_μ :

While the structure is not fully developed yet, the logic is clear. Gravitation is centralized and symmetric. Mass curves space in a spherical shell around it. The electromagnetic wave, the photon, is antisymmetric spinning spacetime. The electron (charge) is centralized spin. Antisymmetric curvature is a matter for study.

TYPE                                 POTENTIAL FIELD

Gravitational curvature
Symmetric = Centralized                q^a_μ^(S)
Einstein gravitation

Gravitational curvature
Antisymmetric = turning
Unexplored. The strong                 q^a_μ^(A)
field, dark matter?

Electromagnetic Field
Antisymmetric EM                 A^a_μ^(A) = A⁽⁰⁾ q^a_μ^(A)
Photon, EM waves

Electrodynamics
Symmetric EM                       A^a_μ^(S) = A⁽⁰⁾ q^a_μ^(S)
Charge, the electron

p.184

We have seen that G q^a_μ = kT q^a_μ. This says that curvature, G, and stress energy, T, are related through the tetrad matrix. The unified field is the tetrad q^a_μ.

That's intellectual fraud: Due to EINSTEIN we know the equation G = kT. To multiply that by the factor q^a_μ is trivial. And saying that both G and T are related by q^a_μ is rubbish! G and T were related by Einstein without Evans' factor q^a_μ.

The tetrad is an asymmetric square matrix. This can be broken into its symmetric and antisymmetric parts:

q^a_μ = q^a_μ^(S) + q^a_μ^(A)                                                 (3)

The splitting is coordinate dependent, hence "physically not meaningful".

p.296

Basis vectors

Unit vectors e₍₁₎, e₍₂₎, e₍₃₎, e₍₄₎ which define a mathematical tangent space.

Read "Unit vectors e₀, e₁, e₂, e₃ ...". The brackets at the indices on p.296-297 should be removed for sake of consistency.

The basis vectors of a reference frame are a group of four mutually orthogonal vectors. Basis vectors establish a unit vector length that can be used to determine the lengths of other vectors. They obey:

e₀² + e₁² + e₂² + e₃² = +1

Due to the following equations we have e₀² + e₁² + e₂² + e₃² = −1 + 1 + 1 + 1 = 2 .

That is, they form a four dimensional sphere. In addition:

e₀·e₀ = −1 ,   e₁·e₁ = e₂·e₂ = e₃·e₀ = +1

e_a·e_b = 0 if a ≠ b; a, b are 0, 1, 2, or 3.

Following that question we shall recognize that the parameter representation R = R(u₁,u₂,u₃) is not sufficient to represent the spacetime manifold under consideration.

p.297

Orthonormal means they are both orthogonal and normalized. If one finds the orthonormal vectors at an event point, then one can define the vectors at any other event point in the spacetime. The spacetime is curved and some method of calculating and visualizing the spacetime is needed. Unit vectors are tangent to a curve at a point. For example the three unit vectors of a curved coordinate system are mutually orthogonal ("perpendicular") and cyclically symmetric with O(3) symmetry. With e_(1,2,3) the unit basis vectors and u_(1,2,3) the coordinates at any point.

The three coordinates u₁, u₂, u₃ can parametrize a three-dimensional manifold only by R = R(u₁,u₂,u₃). However, spacetime has four dimensions. Below only three tangential unit vectors e₁, e₂, e₃ appear.

e₍₁₎·e₍₂₎ = 0, e₍₁₎·e₍₃₎ = 0 and e₍₂₎·e₍₃₎ = 0

and

e₍₁₎×e₍₂₎ = e₍₃₎, e₍₁₎×e₍₃₎ = e₍₂₎ and e₍₂₎×e₍₃₎ = e₍₁₎

The vectors e_(k) are 4-vectors. And the operation × is not defined in 4-D. cf. p.341 Cross product.

e_(n) = 1/h_i × ∂R/∂u_i     read e_i = 1/h_i × ∂R/∂u_i

and the arc length is:

ds = |dR| = |∂R/∂u₁×du₁ + ∂R/∂u₂×du₂ + ∂R/∂u₃×du₃|

Here the operators × should be removed as being no cross product.

p.329

M(A) = M(S) + M(A-S)

Asymmetric equals Symmetric plus Anti-Symmetric

The splitting above from M(A) = M(S) + M(A-S) is no splitting into symmetric and antisymmetric parts.

p.331

Metric tensor

ds² = g_μν x^μ x^ν = q^μq^ν

read ds² = g_μν dx^μdx^ν (= q^μq^ν ?).

The latter part in (...) is evidently wrong as it depends on the summation indices μ and ν. Probably the author has meant the equation

ds² = g_μν dx^μdx^ν = q_μq_ν dx^μdx^ν .

However, the last equation would imply that the matrix must have a vanishing determinant, because all line vectors are parallel to the line [q₀ q₁ q₂ q₃].

The metric tensor is defined as the outer product of two metric vectors. Form the outer product of two four vectors, i.e. multiply a column four vector by a row four vector, and you have a 4×4 matrix, with sixteen components.

However, in that case we obtain det(g_μν) = 0 due to the above argument.

. . . The symmetric metric is always defined as the tensor or outer product of two vectors.

We have quite generally det (A^TÄB) = 0 for two arbitrary n-vectors A, B and their outer product A^TÄB. However, the matrix g = det(g_μν) is not allowed to vanish since the metric matrix must be invertible. Thus:

The symmetric metric (g_μν) cannot be defined as outer product of two 4-vectors.

p.332

We want a real number to define the distance. The metric tensor takes two vectors which define the curvature that occurs and calculates the real distance between them. We cannot have a negative distance so the metric must be "positive definite" – it must be real.

WRONG! Even the special metric of Minkowkian spacetime (cf. p.333), the matrix (η_μν), is indefinite, i.e. ds² = − c²dt² + dx² + dy² + dz² attains negative values inside the light cone and positive values outside while it vanishes at the cone itself. And the determinant of the Minkowskian metric, det(η_μν) has the value −1. Thus:

The Minkowskian metric (η_μν) is a counter example of a metric matrix that cannot be written as an outer product of two 4-vectors.

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